Calculus Examples

$x^{5} \sin (x) \cos (x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{5} \sin (x)$ and $g (x) = \cos (x)$ .

$x^{5} \sin (x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x^{5} \sin (x)]$

Step 2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$x^{5} \sin (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [x^{5} \sin (x)]$

Step 3

Raise $\sin (x)$ to the power of $1$ .

$x^{5} (- (\sin^{1} (x) \sin (x))) + \cos (x) \frac{d}{d x} [x^{5} \sin (x)]$

Step 4

Raise $\sin (x)$ to the power of $1$ .

$x^{5} (- (\sin^{1} (x) \sin^{1} (x))) + \cos (x) \frac{d}{d x} [x^{5} \sin (x)]$

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{5} (- {\sin (x)}^{1 + 1}) + \cos (x) \frac{d}{d x} [x^{5} \sin (x)]$

Step 6

Add $1$ and $1$ .

$x^{5} (- \sin^{2} (x)) + \cos (x) \frac{d}{d x} [x^{5} \sin (x)]$

Step 7

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{5}$ and $g (x) = \sin (x)$ .

$x^{5} (- \sin^{2} (x)) + \cos (x) (x^{5} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x^{5}])$

Step 8

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$x^{5} (- \sin^{2} (x)) + \cos (x) (x^{5} \cos (x) + \sin (x) \frac{d}{d x} [x^{5}])$

Step 9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$x^{5} (- \sin^{2} (x)) + \cos (x) (x^{5} \cos (x) + \sin (x) (5 x^{4}))$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$x^{5} (- \sin^{2} (x)) + \cos (x) (x^{5} \cos (x)) + \cos (x) (\sin (x) (5 x^{4}))$

Step 10.2

Combine terms.

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Step 10.2.1

Raise $\cos (x)$ to the power of $1$ .

$x^{5} (- \sin^{2} (x)) + x^{5} (\cos^{1} (x) \cos (x)) + \cos (x) (\sin (x) (5 x^{4}))$

Step 10.2.2

Raise $\cos (x)$ to the power of $1$ .

$x^{5} (- \sin^{2} (x)) + x^{5} (\cos^{1} (x) \cos^{1} (x)) + \cos (x) (\sin (x) (5 x^{4}))$

Step 10.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{5} (- \sin^{2} (x)) + x^{5} {\cos (x)}^{1 + 1} + \cos (x) (\sin (x) (5 x^{4}))$

Step 10.2.4

Add $1$ and $1$ .

$x^{5} (- \sin^{2} (x)) + x^{5} \cos^{2} (x) + \cos (x) \sin (x) (5 x^{4})$

Step 10.3

Reorder terms.

$- x^{5} \sin^{2} (x) + x^{5} \cos^{2} (x) + 5 x^{4} \cos (x) \sin (x)$